In a plurality of applications in technical systems, parts or components can be subject to stresses, both mechanical and thermal, which alternate or vary over time. In such cases individual parts can for example be subject to direct mechanical stresses through the occurrence of compressive or tensile forces. A time-varying thermal stress of this type arises on the other hand for example for the parts or components in a turbine system, especially in a gas turbine, when the gas turbine is started up or shut down.
Extreme cyclic loading, both mechanical and thermal, results in material fatigue, which, in many cases limits the life of the component. In general, fatigue crack growth (FCG) under cyclic loading is oftentimes the life limiting mechanism of a component. Small cracks typically nucleate from inherent flaws, such as preexisting flaws in a forging, or initiate by other crack initiation mechanisms such as Low Cycle Fatigue (LCF) when a component is in service. During each cycle, typically a start and shutdown of an engine or another change of operating condition, small cracks grow incrementally without impacting the structural integrity of the component. This is referred to as stable crack growth. When the crack sizes reaches a critical size, the crack growth becomes unstable and the component fails. The cycle N at which this happens is called the fatigue crack life of the component. The crack growth can be estimated by Linear Elastic Fracture Mechanics (LEFM) in conjunction with the appropriate transient stress field of the component estimated by Finite Element Analyses (FEA).
In LEFM, the material properties that are of utmost importance for estimating the fracture mechanics life are the fracture toughness (K1c(T)) and the crack growth rate (da/dN(ΔK, T, R)), where da=crack size increment, dN=cycle increment, ΔK=stress intensity factor difference characterizing the stress cycle, T=Temperature, R=R-ratio[stress minimum/stress maximum]. Extensions of LEFM that take into account non-linear effects, additionally require tensile properties, such as yield strength (RP02), ultimate yield strength (RM), and Young's modulus (E). Known examples of theses extensions are Failure Assessment Diagram (FAD) and Irwin Plastic Zone Extension (IPZE).
Furthermore, initial flaw sizes need to be known in order to perform a LEFM calculation. Once these parameters are known, the fatigue crack life N can be calculated by approximating the crack by a plate solution of, for example, an elliptical embedded or semi-elliptical surface crack.
Measurements show that all of those material properties and initial flaw sizes have inherent scatter due to the complexity of manufacturing parts. Furthermore, aging of material properties depend strongly on the individual component operation, initial material microstructure, and chemical composition and therefore lead to a different scatter in the material properties of the aged component.
Due to the uncertainties in material properties and initial flaw sizes, the estimation of fracture mechanics life is complicated. Therefore, for many components fatigue crack growth is not even designed for, or extremely conservative assumptions are made. This can lead to a very conservative design which does not use all the potential of a component.
Two state of the art approaches of fatigue crack life calculation will now be described.
The first approach involves deterministic fracture mechanics life calculation based on minimum/maximum material properties and flaw-sizes. This approach falls under the so-called safe-life design philosophy and is primarily used for land-based heavy duty gas and steam turbines. In this approach, conservative estimates of material properties and initial flaw sizes are used. For example, the fracture toughness (K1c) is estimated by a minimum curve with most of the measured data having larger K1c values. For the crack growth rate (da/dN) a maximum curve is estimated with most of the measured data having smaller values. The flaw-size is typically estimated by the resolution of the non-destructive examination (NDE) technique that is performed before the component is used in an engine or during a life time extension (LTE).
With the above-described ‘worst case’ assumptions, the fatigue crack life N can be conservatively estimated by LEFM, or the aforementioned extensions such as FAD and IPZE analysis for one location of the component. The location provides the stress/temperature input for the LEFM analysis. This location, or sometimes multiple locations, is chosen in such as way that it is the life limiting location, i.e. having the largest stress values and amplitude. The cycle number N at which the crack growth becomes unstable is then assumed to be the fatigue crack life of the component. Sometimes yet another safety factor is applied either on N or elsewhere.
The drawback of such a deterministic fracture mechanics calculation is that the quality of a component is solely based upon a single or only a few locations of the component and minimum/maximum material properties. The information about material data and flaw size scatter is not used at all. Furthermore, due to the scatter of input parameters there will still be a very low probability of failure (PoF) that the component has for the calculated number of cycles N. The PoF is not determined by the deterministic method as it uses only minimum/maximum property/flaw size estimations.
The second approach involves probabilistic fracture mechanics calculation based on flaw-size distribution and distributions of inspection intervals. This particularly used in the aero engine industry. To this end, a tool DARWIN™ (Design Assessment of Reliability with Inspection) has been developed that calculates the accumulated probability of failure (PoF) as a function of cycles N. This methodology focuses on the influence of inspection intervals, initial flaw size distribution, and the mixing of different missions such as start, climbing, cruise at different altitude, in-flight changes in altitude, go-arounds etc. This approach does not account for material property variations. The user can however add a width of the life distribution <N>. This width of the life distribution can be estimated by a series of complex testing that compounds all individual material scatter under different loading conditions or the width is deemed not to be important and set to 0. The user then has to define a limited number of spatial zones within the Finite Element Model (FEM). In each of those zones a crack is defined and a limited number of LEFM fatigue crack growth calculation are performed that represent the possible flaw size distribution. Each zone i is thereby assigned with a probability of failure PoF (N,i), the total PoF (N) is then calculated based on a complex summation of the individual PoF (N,i). The total number of LEFM calculations performed is limited as the problem gets computationally involved and execution times increase. For low PoF this can lead to errors in the estimation of the PoF as details of the failure surface might not be correctly resolved.
The drawback of the second approach is that it does not account for individual material scatter and evaluates the risk for each zone by one representative crack, i.e. the user puts the crack in a location in each zone that he thinks is representative for the whole zone. Therefore, a zone-refinement convergence check has to be performed to see whether the results still depend on the zone size.